
Again, this equation corresponds to Eq.(26) in [1]} but distinct from it in the sense that it incorporates an additional terms (second term on the left hand side and last term on the right hand side) emerging due to the curvature emendation. This when integrated along with the initial assumption \(\Pi (t=0)=0\)  produces the following outcome
\(\Pi =-\frac{f_R}{4\pi \hat{R}}\int _0^t \hat{R}\left(\dot{\epsilon }+\frac{3\epsilon \dot{\hat{R}}}{\hat{R}}\right)d\tilde{t}-\frac{f_R}{\hat{R}}\int _0^t\frac{(\mu +P_r)A\sigma \hat{R}}{f_R}d\tilde{t}+\frac{f_R}{4\pi \hat{R}}\int _0^t\hat{R}(\Psi _{diss}+\vartheta _1)d\tilde{t},\) 
